How to predict time series with accuracy?
Things like this a really hot topic in research right now, and it's very difficult to get high accuracy on a chaotic system like the stock market. That being said, I would probably recommend preprocessing your data rather than having your primary neural network decide what to accept and what not to. 

For example, in your specific case, you could model a bubble bursting as perhaps a negative exponential drop-off or something of the sort. This could include machine learning too. You could gather historical drops in stock market data, and use some sort of regression (Bayesian would probably work well) to estimate the best function to use as an indicator to whether a steep drop has occurred. If so, then use your neural network specifically to classify the fate of the stock. I would think you would have more success following a specialised route such as this rather than trying to train a network on general trends in the market.

In terms of the structure of your neural network, you may want to consider a convolutional neural network (CNN) instead of a recurrent neural network (RNN). RNNs assume the current point in your time-series depends on all previous points, from the beginning of your data. I wouldn't think this would hold true in general for the stock market. The filters a CNN learns are suited to learning to extract certain features and the CNN will apply the filters to specific portions of the data, in the way it considers optimal. They are both nonlinear models, but the CNN will be less computationally costly to train. You could also try a gradient-boosting regression approach instead of a neural network. That being said, something like an LSTM RNN (long short-term memory) won't necessarily be bad - just my two cents.